It seems this is a relatively basic question but does the LT4320/LT4321 operate down to 12VAC?
LT4320 can operate down to 12VAC with enough bulk capacitance. See DC1902B for an example. LT4321 is designed for DC applications like Power over Ethernet and has a higher operating voltage.
12VAC (RMS) is roughly 17V peak, so bridge's output capacitance will charge up to 17V without a load. The more load, the more ripple at the output. The more output capacitance, the less ripple. You must make sure that the output voltage stays above 9V to assure that the LT4320 operates normally. The part is not designed to operate below 9V, and low output voltages will cause it to shutdown.